Testing CPT violation, entanglement and gravitational interactions in particle mixing with trapped ions

By analyzing the analogies between the effective system of $N$ spins described by the Ising Hamiltonian and the phenomenon of the self--gravity in mixed particle systems, we show that cooled ions held in a segmented ion trap and exposed to a magnetic field gradient can simulate the proposed mechanism of mutual interaction in mixed neutrino system. We show that with trapped ions one can reproduce the expected corrections to the flavor transitions and the $CPT$ violation induced by gravity on flavor fields, which may have played an important role in the early stages of the universe. The results presented are experimentally testable. They indicate that ions confined in microtraps can represent a new tool to test fundamental phenomena of nature.


I. INTRODUCTION
Particle mixing and oscillations represent the most compelling indication of physics beyond the standard model of particles [1][2][3]. The physical oscillating fields, named flavor fields, are superpositions of free fields with different definite masses. Particle mixing interests axion-photon system [4][5][6], η-η [7], neutral kaons [8], B 0 and D 0 mesons [9], in boson sector; neutrino flavor oscillations [10,11], neutronantineutron oscillations [12], and the quark mixing [13], in fermion sector. Many efforts have been devoted to understanding the origin of the phenomenon and, recently, also the effects of gravity on neutral mixed particles have been analyzed [14][15][16][17]. Moreover, gravity has been also considered as a source of decoherence in flavor oscillations [18][19][20] and the effects of decoherence for mixed particles on fundamental symmetries of nature have been recently studied [21][22][23][24][25][26][27][28]. It has been shown that gravity can affect the frequency of oscillations [16,17] and leads to many interesting effects like the CPT-symmetry violation in particle mixing [21][22][23][24][25][26][27]. Effects of quantum field theory in curved space-time on mixed neutrinos have been also analyzed [29]. The phenomena mentioned above are difficult to reveal experimentally, due to the elusive nature of neutrinos and the short lifetime of composite particles (as pertaining to meson mixing).
Another particularly thriving field of research is represented by the study of trapped ions. Trapped ions are among the most promising systems for practical quantum computing and many quantum technological applications have been developed by using such systems. An ion trap is a combination of electric or magnetic fields used to capture ions realizing a system well isolated from the external environment. Therefore it is characterized by long lifetimes and coherence [30,31], strong ion-ion interactions [32], and the existence of cycling transitions between internal states of ions that allows to realize both measurement and laser cooling. Trapped ions, besides being one of the leading technology platforms for large-scale quantum computing, allow the simulation of quantum models of strongly interacting quantum matter and enable the investigation of those quantum dynamics that remain so far unexplored due their inescapable complexity [33,34].
In this work, we show that trapped ions can also simulate the gravity effect on mixed neutral particles and thus reproduce, in table top experiments, analogues of phenomena that may have characterized the first stages of the evolution of the Universe.
In particular, we analyze the analogies between a selfinteracting system of mixing particles and an ensemble of Doppler cooled ions loaded in a magnetic trap. We note that an effective spin-1/2 long range Ising model can describe both Doppler cooled ions held in a segmented ion trap [32,[35][36][37][38] and the dynamics of a gravitationally self-interacting system made of oscillating neutrinos, as introduced in Ref. [17]. In that paper, it has been studied an ensemble of N mixed selfinteracting neutral particles as a closed system and it has been shown that the self-gravity effects induce a CP T violation and a modification of the oscillation formulae for systems like neutrinos. The emerging entanglement and the CP T violation are extremely small for systems like neutrinos or neutrons, yet such effects might play a crucial role in dense matter and during the early stages of the universe.
Here, we consider different interaction potentials and we analyze the oscillation formulae and the T -symmetry violation for traps loaded with 4 or 6 ions. We start by analyzing the case with 4 ions and coupling in three independent wells, in which the inner ions are confined in a common well and coupled strongly. Then we consider 6 ions confined in a single strongly an-harmonic well. We show that trapped ions reproduce the expected corrections to the flavor transitions and the CP T violation induced by gravity on flavor fields. We show that the T violation for confined ions is the analogue of the CP T violation for flavor mixed fields and we find that T violation depends on the number of trapped ions, as predicted for CP T violation in the flavor mixing case. Our numerical results show that such violations, together with the new oscillation formulae are all detectable in the present experiments on ion traps. Reproducing the effects predicted for neutrinos in atomic systems could open new scenarios for the study of fundamental physics; it could be of high relevance in the understanding of the evolution of the universe and possibly lead to the discovery of new phenomena.

arXiv:2101.06469v1 [hep-ph] 16 Jan 2021
The paper is organized as follows. In Sec. II, for the reader's convenience, we briefly resume the main results of the treatment of gravitationally self-interacting oscillating neutrinos. In Sec. III, we present the analogies with cooled trapped ions and we show that the very tiny effects induced by the gravitational self-interaction for neutrinos have corresponding, larger effects in trapped ions, which can be easily detected. Indeed, we show that the oscillation formulae for trapped ions and the T symmetry violation which can be interpreted as a signature of CP T violation in mixed flavor fields, can be experimentally tested with the present technologies. In Sec. VI we draw our conclusions.

II. OSCILLATIONS OF N INTERACTING PARTICLES
In this section we report the main results obtained for oscillating and gravitationally interacting particles [17]. The mixing relations for two flavor fields n A and n B are where n A = ν e and n B = ν µ for neutrinos, θ is the mixing angle, φ is the Majorana phase which is zero in the case of Dirac fermions [2] and |m i are the states with definite masses m i .
Denoting with E the energy of the mixed particles travelling through the space, and assuming m i E, the Hamiltonian of mixed fields can be written as where σ z = |m 1 m 1 |−|m 2 m 2 |, and we neglected the terms proportional to the identity. Considering the validity of the equivalence principle between inertial and gravitational mass, and representing the gravitational interaction with the Newtonian potential, the Hamiltonian of N mixed particles interacting gravitationally becomes where , Ω i,i = 0 and d i,j relative distance between the i-th and the j-th fields that, for sake of simplicity is assumed constant during the evolution.
Assuming that at time t = 0 the system is described by a fully separable state and that the first M particles are created in the state |n A and the rest is in the state |n B , then the initial state is For t > 0, since the system is closed, one has the pure state |ψ (N ) (t) = U (t) |ψ (N ) (0) , where the unitary evolution operator is U (t) = exp(−ıtH (N ) ). The reduced density matrix on the selected k-th particle is where the index α runs over the ensemble {x, y, z}, the function ζ k is equal to +1 for k ≤ M , and to −1 for k > M , and a k (t) is given by Considering two copies of the N -particle system such that in the first copy one has M = N and in the second one has M = 0, the oscillation probabilities (obtained considering the average over all the elements of the system) are Assuming Ω i,j t 1 the expression of a k (t) simplifies and becomes equal to a k (t) e ıω k t 1 ± 2ı cos(2θ) N j=1 Ω k,j t , where the sign + is for a  sin(2ω k t)Ω k,j .
Being ∆ CP = ∆ T , also the CP T symmetry is violated. Denoting with F = 1 N N k=1 f k the average of f k , which are defined as f k = sin(2ω k t) N j Ω k,j , one can express ∆T in terms of average values of relative distances among the particles in the system as This relation shows the explicit dependence of ∆ T on the number of particles of the system. Similar results are obtained for all the configurations in which, n A − n B ∼ N at t = 0. If n A ∼ n B at t = 0, one has ∆ T ∝ √ N .

III. TRAPPED IONS AS GRAVITATIONALLY INTERACTING PARTICLES
Let us now consider cooled ions held in a segmented ion trap and exposed to a magnetic field gradient in order to real-ize effective spin 1/2 models. The effective spin-spin interactions induced by the magnetic field are of Ising type and can be adjusted by tailoring the axial trapping potential. In particular, if the ions are sufficiently cold, such that the ion motion can be neglected, the effective system of N spins is described by the Ising Hamiltonian [39] H (z) where ω i are the resonance frequencies of the atomic spins, depending on the external magnetic field B (x 0,j ) at the equilibrium position of the ion x 0,j . The spin-spin couplings J ij depend on the trapping potential and on the spatial derivative of the spin resonance frequency, that is determined by the magnetic field gradient. They are given by where A is the Hessian matrix of the potential energy function that confines the ions in the position x j . In addition, the magnetic gradient allows for the addressing of individual spins with a microwave field, which can be used to manipulate the spin dynamics. By comparing the Hamiltonians (3) and (8), we see that they are formally identical, with the obvious correspondence ω i ↔ ωi 2 , J ij ↔ −Ω i,j . Hence the resonance frequencies of the atomic spins ω i play the role of ω i in Eq.(3), apart from a factor 1/2, and the spin-spin couplings J ij correspond to the gravitational couplings Ω i,j except for a sign. Thus the results presented for mixed particle systems can be reproduced with trapped ions.
It is clear, however, that the analogy fails when the antiparticles are involved. This is because one does not have the CP conjugate of the system at his disposal in the case of trapped ions. If one could hypothetically reproduce CP conjugate of the system, and the only interaction within the system were the one of Eq. (8), all the results, including the CP symmetry would be recovered. In any case, since the oscillation probability are not dependent on the CP violating Majorana phase, there is no reason to believe that the CP conjugate oscillations differ from Eq. (5). Then we can infer that the trapped ion system is also CP invariant. In this sense, we can assume that ∆ CP = 0. Since the T violation can be tested in this setup, and in the neutrino mixing case it induces a CP T symmetry violation, trapped ions can be used to indirectly test the CP T violation predicted for neutrinos gravitationally interacting.
We now proceed with a numerical analysis, considering different potential shapes, and we analyze the oscillation formulae and the T violation resulting for the cases in which the trap is loaded with 4 or 6 ions. The initial state of the ion ensemble is assumed to be factorizable as the direct product of single ion states. We consider two possible initial single ion states, written as superposition of spin up |↑ and down |↓ : Notice that both the states of Eq. (11) are pure. We start by studying three different wells loaded with 4 strongly coupled ions. These simulate a non-uniform distribution of neutrinos in space. To investigate a realistic experimental set-up of this configuration, we use the following values of resonance frequencies of the atomic spins splitting: ∆ω i /2π(MHz) = −30.7; −2.3; 2.4; 30.9 , and the following values of the coupling constant J ij ( Hz): J 1,2 = 2.1; J 1,3 = 1.8; J 1,4 = 0.4; J 2,3 = 123.8, with J 3,4 = J 2,4 [34].
Plots of the transition probability P B→A as a function of time are shown in Figs.(1) and the results for the violation of the T -symmetry ∆ T as a function of time for such a system are reported in Fig. 2. For the oscillation probabilities ( Fig. 1) we use the time range [10 −7 − 10 −6 ] s, while the Tasymmetries are plotted in the range [10 −6 − 10 −4 ] s because the magnitude of the oscillations is larger here, and therefore they are more visible.
The plots show that, apart from specific singular values, e.g. θ = π/4, the violation of the T -symmetry, and then of the CP T symmetry, is large enough to be detected with the current technologies. Notice that the time intervals considered (t ∈ [10 −7 − 10 −4] seconds), are well below the coherence time usually characterizing the experiments with trapped ions.
With today's technologies of trapped ions we can also simulate larger systems. However an increment of the number It should be noted that the general prediction of Eq. (7) of the dependence of the T asymmetry on the total particle number N is confirmed by the plots shown above. The 6 ions asymmetry of fig. 4 (given the different couplings) at t 1 = 10 −6 s is slightly bigger than the 4 ions asymmetry of fig. 2 at the same time t 1 and, due to Eq. (7), we expect that a larger number of ions implies a larger T asymmetry. In any case, a limited number of ions is to be preferred, since despite a larger signal, the coherence time is reduced when a large number of ions is considered. The plots obtained show that the oscillation formulae and the CP T violation predicted for gravitationally interacting neutrinos can be tested with ions trapped in many potentials.

IV. CONCLUSIONS
The gravitational self-interaction in particle mixing systems leads to new oscillation formulae and to the CP T violation. The corrections to the flavor transitions and the expected CP T symmetry breaking are negligible in the present epoch and therefore they are very hard to be revealed. However, they are proportional to the number of elements of the system and to its density, so that these effects could affect extremely dense systems with a large number of particles, such as some galactic objects and the primordial stages of the universe. Moreover, the mechanism leading to CP T violation is not only limited to gravitational interaction. Therefore, the possibility of testing these processes is of great importance for the understanding of fundamental physics.
In our work, we have shown the analogies between the effective system of N spins described by the Ising Hamiltonian and the phenomenon of the gravitational self interaction in mixed particle systems. We have shown that cooled ions held in many potential traps can simulate the mutual interaction in mixed neutrino system and allow to reveal the expected new oscillation formulae and the CP T violation for this phenomenon. We considered the case with N = 4 ions and coupling in three independent wells, and the case of 6 ions confined in a single strongly an-harmonic well. We have shown that the T violation for trapped ions, corresponding to the CP T violation for flavor mixed particles, grows with the size of the system N . Our numerical results show that such violations, together with the new oscillation formulae are all detectable in the present experiments on ion traps. Finally, note that in our approach we have chosen the couplings J ij and the frequenciesω i in order to maximize the T -violation. By lowering the interaction, and therefore the couplings and the frequencies, it is possible to slow down and highlight the oscillations. The potentials can always be tuned according to whether one is interested in highlighting the T asymmetry or the oscillation formulae.